Integral transforms Hankel transform

The most common technique for the derivation of fundamental solutions is to use integral transforms, such as, Fourier, Laplace or Hankel transforms [29, 39]. For simple operators, such as the Laplacian, direct integration and the use of the properties of the Dirac delta are typically used to construct the fundamental solution. For the case of a two-dimensional Laplace equation we can use a two-dimensional Fourier transform, F, to get the fundamental solution as follows,... 

GENERALIZED INTEGRAL TRANSFORMATIONS, A.H. Zemanian. Graduate-level study of recent generalizations of the Laplace, Mellin, Hankel, K. Weierstrass, convolution and other simple transformations. Bibliography. 320pp. 5H x 8H. 65375-7 Pa. 7.95... 

Numerically, the Hankel transforms are best obtained by first calculating integral transforms, which we denote by c" " (/-)[or rf"" r) or h "" (r)]. The c" (r) was introduced by Wertheim, and generalized results have been obtained by Blum. Blum shows that the calculation of c " k) [defined by (3.1 Ik)] reduces to taking the zeroth-order Hankel (Fourier) transform of c"" (r) if / is even, or the first-order Hankel transform if / is odd. Thus one has... 

In mathematics, the Hankel transform (Goodman 2005 Bracewell and Bracewell 1986) of order zero is an integral transform equivalent to a two-dimensional Fourier transform with aradially symmetric integral kernel. It is also called theFourier-Bessel transform. 

In this section, we will apply the finite integral transform to a general Sturm-Liouville system, and the integral transform is therefore called the Sturm-Liouville integral transform. Thus, all finite integral transforms are covered at once Fourier, Hankel, and so forth. 

The second integral is the Hankel transform of the Bessel operator of zero order. Using the property of the Hankel transform, Equation (2.248) is transformed with the conditions stated by Eqnation (2.249) to... 

The upper cutoffs are R = N Ar and K = N Ak for the integrals in Equations 6.29 and 6.30, respectively. Note that is not the sampling limit set by the simulation box dimensions, but is typically much larger. The TCFs, DCFs, and their Hankel transforms at an iteration step t are represented by discrete vectors. 

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